Hall effect measurements in YBCO

Badoux, Proust, Taillefer et al., Nature 531, 210 (2016)

14

0 0.1 0.2 0.3p

0

0.5

1

1.5

n H=

V /

e R

H

p

1 + p

SDW CDW FL

p*

a

b

Evidence for FL* metal with Fermi surface of size p ?!

§ Hall conductivity provides strong evidence of Fermi surface reconstruction near optimal doping

▪ Is there a quantum critical point under the superconducting dome?

▪ What is the nature of phase transition?Symmetry breaking or topological?

Phys. Rev. B 94, 115147 (2016), Phys. Rev. Lett. 119, 227007 (2017)

Quantum fluctuating antiferromagnetism: A route to intertwining topological order & discrete broken symmetries

Pseudogap phase in high Tc cuprates

Shubhayu Chatterjee1, Subir Sachdev1,2 and Mathias S. Scheurer1

1Harvard University, USA, 2Perimeter Institute, Canada

Pseudogap Metal:

Displays Fermi arcs.

Behaves like a Fermi liquid, with a Fermi surface of size pinstead of 1 + p.

Hall effect experiments show that it is also

present at high magnetic fields and low

temperatures.

FL

Figure credits: K. Fujita et al, Nature Physics 12,150–156 (2016)M. Plate et al, PRL. 95, 077001 (2005)

Conventional Fermi liquid:

Large hole Fermi surface of size 1 + p.

Possibility 1: Symmetry breaking: Spin density wave

(SDW) order

Possibility 2: Topological order (No long range symmetry breaking)

M. Oshikawa, PRL 84, 3370 (2000)T. Senthil et al, PRL 90, 216403 (2003)Paramekanti et al, PRB 70, 245118 (2004)

Z2 topological order in metals

Sachdev et al, PRB 80, 155129 (2009)

Spin fermion model: Electrons coupled to O(3) AF order parameter

Transform to ‘rotating reference frame’ defined by local orientation of the O(3) order parameter

Fermionic chargons

Higgs fieldHiggs field = AFM order for the chargons

Chowdhury et al, PRB 91, 115123 (2015)Sachdev et al, PTEP 12C102 (2016)

• SU(2) gauge theory of metals with Z2 topological order can explain the concurrent appearance of anti-nodal gap and discrete broken symmetries in the hole-doped cuprates.

• Topologically ordered phases energetically proximate to the Neel state have the desired broken symmetries.

• How does one relate the parameters of the theory to the microscopic hopping/interaction parameters measured in experiments?

• What are the signatures of topological order in numerics, like cluster DMFT on the 2d Hubbard model?

• Is time-reversal symmetry broken in the hole-doped cuprates? If not, how does one get a topological metal with broken inversion inversion but intact time-reversal?

• What about phase transitions to superconductivity/density wave-phases?

Additional broken symmetries in the pseudogap phase

Can discrete symmetry breaking be intertwined with topological order?

§ Nematic order, broken C4 symmetryDaou et al, Nature 463, 519 (2010)

§ Broken inversion symmetry C2

§ Broken time reversal 𝜭 (?)

§ 𝜭 C2 seems to be preserved Zhao,Belvin,Hsiehetal,NaturePhysics13, 250 (2017)

Do such phases appear naturally proximate to a Neel antiferromagnet?

Long range AF order close to Neel phase

0.8 0.9 1.0 1.1- 0.3

- 0.2

- 0.1

0.0

0.1

0.2

0.3

Phases of classical frustrated Heinsenberg model with ring exchange K Identical SDW phases in metal

Z2 topological order in insulators

Key idea: Quantum disorder the spins: Spin-rotation and translation invariance regained. Discrete symmetries remain broken.

Conclusions and open questions

Odd under C2 and 𝜭, but even under 𝜭 C2– same symmetries as loop currents

Condense charge 2 Higgs fields

CP1 model with emergent U(1) gauge field

For conical spiral H, this phase is a metal with:1. Anti-nodal spectral gap (small FS)

2. Loop current order

Topological metal violates Luttinger’s Theorem